We present some models of viscoelastic systems with long-memory behaviour; we then provide both thorough mathematical analysis and numerical simulations of these models by means of so-called diiusive representations. The models under consideration are the basic harmonic oscillator with fractional damping, the generalized Lokshin model of waves, and also the boundary feedback control of the Euler-Bernoulli beam by impedance matching, which involves fractional integro-diierential operators. We use original diiusive input-output representations of the fractional damping operators, which rst permits to establish the positivity of the damping, second to interpret the viscoelastically damped models as coupled problems between a standard system and a diiusion equation (devoted to damping). This approach naturally introduces a global energy functional, the evolution of which may then be analysed from standard Lyapunov and LaSalle techniques. Some relevant numerical simulations are nally presented in order to illustrate the approach.
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